To what extent does the parallelogram law hold?
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To gain a deep understanding of these questions, still today the subject of very active research, powerful techniques were developed by leveraging surprising connections between geometry, analysis and probability. These have now found numerous applications across mathematics and computer science.
In particular, a rich interplay of continuous and discrete mathematics has developed in recent years. Related content: Vershynin's Chapter 1 link. A lot of effort goes into pointing out the similarities and the differences.
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Since also Lebesgue integrals are used, an appendix introduces this concept in the one dimensional case. But friendly does not mean that the reader is spared from effort. The text is peppered with many exercises.
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The formulation of the exercises and the solutions provided at the end form an essential part of the book and they actually fill the larger part of its pages. Thus the reader who is new to the topic is supposed to work hard to properly grasp all the ideas. By including the solutions, the book is suitable for self study, although the text grew out of lecture notes used by the author while teaching at the London School of Economics. On the other hand, the subject is vast, and since this is only an introduction not all the most difficult proofs are provided in full extent and not all the most complicated issues are discussed in all detail.
It is important to mentioned that, even though the subject is abstract, attention is also paid to the application aspect. In fact the introduction starts with an optimal control problem as a motivation to embark on functional analysis.
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In a later chapter when differentiation and its application to optimality conditions is discussed, the Euler-Lagrange equation is derived and it is applied to classical Hamiltonian and Poissonian mechanics. This forms also the basis for a discussion of quantum mechanics in the chapter on Hilbert spaces. Compact operators are a reason to go into the subject of finite dimensional Galerkin approximations of the operators, which is important for numerical computations. Weak solutions of differential equations are obtained using distributions, discussed in the last chapter.
Topics in Real and Functional Analysis
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Active 2 years, 1 month ago. Viewed times. I require lecture notes for self studying purpose. Added: How is the Monograph? Smyrlis Jul 3 '17 at Yiorgos S.